You want a "satisfactory answer," but do not indicate what would satisfy you. Thus, I am going to stab a guess and take a very different approach from the other answer and comments. I don't know any strict grammar rules but will go by what naturally occurs to me as a native English-speaker and also native Pennsylvania German speaker, both languages of which use the same rules when talking about wind or a wind.
It depends on the situation regarding whether one says "a wind" or "the wind" or just "wind." You provide three examples:
- There’s a wind blowing through here tonight.
- A wind is a current of air that is moving across the earth's surface.
- Open the doors, I want to get a breeze.
I'll omit Sample 2 because my scientific knowledge is not good enough to deal with it. Samples 1 and 3 are very familiar situations in my life so I'll deal with them.
There’s a wind blowing through here tonight.
That's just the way we say it. One might also say, "There's wind blowing through here tonight." Or "The wind is blowing through here tonight." I don't think one is more correct than the other.
Let's discuss which would be more likely to be used when. I will envision a hot muggy summer evening and a group of friends sitting in lawn chairs in someone's backyard chatting about whatever is on their minds when someone observes, "There's a wind blowing through here tonight."
It's just a natural normal wind that cools the air and everyone is grateful for it.
Now let's imagine that instead, the person had observed that, "The wind is blowing through here."
That's still the natural normal wind that cools the air and everyone is grateful for it, but they are surprised that it blows through there. They had imagined that the location was too sheltered to get much breeze or wind. But it wasn't. Nice surprise.
But the other one, "There's wind blowing through here tonight" might mean puzzlement. Is there something wrong that causes wind to blow through here? For example, maybe there's a huge generator fan with an open door or window where there shouldn't be. That would be a different kind of wind from the natural normal wind discussed above. Might be time for someone to check things out.
Then again, if nothing appears to be ammiss, possibly that was just this person's way of noting that the wind was blowing through there. I personally don't know of any hard and fast grammar rules about this.
Open the doors, I want to get a breeze.
This time I envision a stuffy indoor situation. It would not sound right to say, "I want to get breeze." That would immediately mark a person as a non-native English-speaker. It would be okay to say, "There's a breeze out there. I can see the trees moving. Open the doors. I want to feel the breeze in here, too." When the doors are open, one might say, "It's breezy in here."
One can also say "Breezes are blowing across the meadows." However, if you want to talk about the summer breezes or the drying breezes, you need the definite article. Also for just one breeze--however that is determined, one must say "a breeze" or "the breeze."
Again, this is just the way we talk. Maybe there are no grammar rules to explain it, except for the singular and plural of breeze(s). I'm not sure where to look.